Moduli of Continuity of Local Times of Strongly Symmetric Markov Processes via Gaussian Processes 1. Introduction. In this paper we show that the moduli of continuity of the local times of a large class of Markov processes can be easily obtained if one knows the moduli of continuity of the Gaussian processes that are associated with the Markov processes. And, since the moduli of continuity results for the Gaussian processes are either well known, or in many cases easy to obtain, we get similar moduli of continuity results for the local times. This paper is an application of [MR] in which we obtained necessary and sufficient conditions for continuity and boundedness of the local times of strongly symmetric standard Markov processes by establishing the equivalence of these conditions with those of the associated Gausssian processess. The key ingredient in this work is an application of Lemma 4.3 of [MR] which is itself a corollary of an isomorphism theorem of Dynkin [D], [D1] which relates local times and Gaussian processes. (For another application of these methods see [MR1]). Let (S, ρ) be a locally compact metric space with a countable base and let X = (Ω, F t ,X t ,P x ), t ∈ R + , be a strongly symmetric standard Markov process with state space S . In saying that X is symmetric we mean that there is a σ–finite measure m( · ) on S such that the Markov transition function P t satisfies (P t f,g)=(f,P t g) ∀t ∈ R + for all measureable functions f and g in L 2 (S ) where (f,g) ≡ f g dm is the usual inner product. In saying that X is strongly symmetric we mean that in addition to X being symmetric the measure U α = U α (x, · ) given by U α (x, · )= ∞ 0 e -αt P t (x, · ) dt is absolutely continuous with respect to m for some α> 0, (and hence for all α> 0). In this case there is a canonical symmetric α–excessive density u α = u α (x, y) for U α . Moreover a strongly symmetric standard Markov process X has a symmetric transition density function p t (x, y) and u α (x, y)= ∞ 0 e -αt p t (x, y) dt Let L = {L y t , (t, y) ∈ R + × S } denote the local times of X . It is known that a necessary and sufficient conditon for the existence of a local time for a strongly symmetric standard 1
Transcript
Moduli of Continuity of Local Times of Strongly
Symmetric Markov Processes via Gaussian Processes
1. Introduction. In this paper we show that the moduli of continuity of the local times
of a large class of Markov processes can be easily obtained if one knows the moduli of
continuity of the Gaussian processes that are associated with the Markov processes. And,
since the moduli of continuity results for the Gaussian processes are either well known,
or in many cases easy to obtain, we get similar moduli of continuity results for the local
times. This paper is an application of [MR] in which we obtained necessary and sufficient
conditions for continuity and boundedness of the local times of strongly symmetric standard
Markov processes by establishing the equivalence of these conditions with those of the
associated Gausssian processess. The key ingredient in this work is an application of
Lemma 4.3 of [MR] which is itself a corollary of an isomorphism theorem of Dynkin [D],
[D1] which relates local times and Gaussian processes. (For another application of these
methods see [MR1]).
Let (S, ρ) be a locally compact metric space with a countable base and let X =
(Ω,Ft, Xt, Px), t ∈ R+, be a strongly symmetric standard Markov process with state
space S. In saying that X is symmetric we mean that there is a σ–finite measure m( · ) on
S such that the Markov transition function Pt satisfies
(Ptf, g) = (f, Pt g) ∀t ∈ R+
for all measureable functions f and g in L2(S) where (f, g) ≡∫fg dm is the usual inner
product. In saying that X is strongly symmetric we mean that in addition to X being
symmetric the measure Uα = Uα(x, · ) given by
Uα(x, · ) =∫ ∞
0
e−αtPt(x, · ) dt
is absolutely continuous with respect to m for some α > 0, (and hence for all α > 0).
In this case there is a canonical symmetric α–excessive density uα = uα(x, y) for Uα.
Moreover a strongly symmetric standard Markov process X has a symmetric transition
density function pt(x, y) and
uα(x, y) =∫ ∞
0
e−αtpt(x, y) dt
Let L = Lyt , (t, y) ∈ R+×S denote the local times of X. It is known that a necessary
and sufficient conditon for the existence of a local time for a strongly symmetric standard1
2
Markov process is that
uα(x, y) <∞ ∀x, y ∈ S
We assume this throughout this paper and normalize the local time by setting
(1.1) Ex
(∫ ∞
0
e−αt dLyt
)= uα(x, y)
(If (1.1) holds for any α > 0 it holds for all α > 0).
It is well known, (see e.g. Theorem 3.3 [MR]), that the function uα(x, y) is positive
definite on S × S for each α > 0. Therefore, for each α > 0, we can define a mean zero
Gaussian process Gα(y), y ∈ S with covarience
E(Gα(x)Gα(y)) = uα(x, y) ∀x, y ∈ S
The processes X and Gα(y), y ∈ S, which we take to be independent, are related through
the α–potential density uα(x, y) and are referred to as associated processes. To simplify
the statement of our results we will always consider X and the associated Gaussian process
corresponding to α = 1, that is G1(y), y ∈ S. In what follows we denote this process by
G = G(y), y ∈ S and note that the natural L2 metric induced by G is a function of the
Proof. In each of the four cases it is straightfoward to verify that ρ satisfies (5.1)–(5.3). We
will show that ψ(u) ≈ ψ(u) where we use ψ(u) ≈ g(u) to mean that there exist constants
0 < c ≤ c′ <∞ such that c ≤ ψ(u)/g(u) ≤ c′, for all u ∈ [0, u0] for some u0 > 0. This gives
(5.5) with ψ replaced by ψ and possibly different constants. Nevertheless, this still implies
(5.4) with ψ replaced by ψ just as in the proof of Theorem 5.1. It is easy to check the
approximation for ψ in (5.6). The approximation for ψ given in (5.7) comes from Lemma
3.9 [M]. The approximations for ψ given in (5.8) and (5.9) can be easily calculated using
the following lemma which expresses Iρ(δ) in terms of an integral that is easier to compute.
Lemma 5.3. Let ρ be a non–decreasing function which satisfies (5.3), let
(5.10) φ(δ) =∫ δ
0
ρ(u)u(log 1/u)1/2
du
and assume that
(5.11) ρ(δ)(log 1/δ)1/2 = O(φ(δ))
Then φ(δ) ≈ Iρ(δ).
Proof. By (5.3) ρ(2u) ≤ 2αρ(u). Using this we see that
φ(δ) ≤ 2αψ(δ)
The other direction follows from Lemma 4.4 [M].
Of course the above results can be applied to real valued Levy processes. In the rest
of this paper we will concentrate on these processes. Let X(t), t ∈ R+ be a real valued
symmetric Levy process, i.e.
(5.12) E exp(iλX(t)) = exp(−tψ(λ))
where
(5.13) ψ(λ) = 2
∞∫0
(1− cosλu)ν(du)
26
for ν a Levy measure. It is known that X has a local time if and only if (1 + ψ(λ))−1 ∈L1(R+), (see e.g. [Br] and [K]). It is easy to see, by considering the characteristic function
of X, that when (1 + ψ(λ))−1 ∈ L1(R+)
(5.14) u1(x, y) + u1(y, y)− 2u1(x, y) =2π
∞∫0
1− cosλ(x− y)1 + ψ(λ)
dλ ≡ σ2(|x− y|)
Thus by (1.2) a symmetric real valued Levy process is associated with a stationary Gaussian
process for which d(x, y) = σ(|x− y|).The follwing Lemma shows that for symmetric Levy processes on the real line, the
function σ2, defined in (5.14), can not be regularly varying with index greater than 1.
Lemma 5.4. Let X(t), t ∈ R+ be a real valued symmetric Levy process and let ψ and
σ be as defined in (5.13) and (5.14). Then
ψ(λ) = O(λ2) as λ→∞(5.15)
and
x = O(σ2(x)) as x→ 0(5.16)
Proof. (5.15) is well known but we include it because it is used for (5.16). We have
ψ(λ) = 2
∞∫0
sin2 λx
2ν(x)
≤ λ2
2
1∫0
x2ν(dx) + 2
∞∫1
ν(dx)
which, using properties of the Levy measure ν, gives (5.15). By (5.15) there exist constants
C and λ0 such that ψ(λ) ≤ Cλ2 for all λ ≥ λ0. Let N ≥ λ0 be such that 1/N2 ≤ C. Then
for x < 1/N we have
σ2(x) = C1
∞∫0
sin2 λx
21
1 + ψ(λ)dλ(5.17)
≥ C2x2
1/x∫N
11/N2 + C
dλ
≥ C2x2
2C
(1x−N
)
27
where C1 and C2 are strictly positive constants. Clearly, (5.17) implies (5.16).
We now present examples in which we can obtain exact moduli of continuity for some
symmetric Levy processes. The next theorem gives an exact local modulus of continuity
and complements a similar result of Barlow, [Ba, Theorem 2], which deals with the exact
uniform modulus of continuity.
Theorem 5.5. Let X be a real valued symmetric Levy process with local time L =
Lyt , (t, y) ∈ R+ × R. Let σ(|u|) be as defined in (5.14), (so that for symmetric Levy
processes σ(|x− y|) = d(x, y)) and let I be a closed interval of R. Then, if σ2 is regularly
varying with index 0 < α ≤ 1, for each y0 ∈ I
(5.18) limδ→0
sup|u|≤δ
y0+u∈I
|Ly0+ut − Ly0
t |(σ2(|u|) log log 1/|u|)1/2
= 2(Ly0t )1/2 for almost all t a.s.
Proof. This result is a direct consequence of Theorem 1.1 in this paper and Theorems 5
and 6 in [Ko] keeping in mind, as we have remarked above, that Theorem 1.1 is also valid
for the modulus in (5.18) and also that ζ = ∞ for Levy processes. (In his Theorems 5
and 6 Kono obtains a version of (1.4) for stationary Gaussian processes with L2 metric
σ2 satisfying the conditions of this Theorem. In fact Kono’s conditions on σ2 is actually
weaker that the one in the statement of Theorem 5.5. In Theorems 5 and 6 Kono adds
the condition that σ2 is concave but he only uses the regular variation of σ2 in the parts
of these Theorems that we are using here).
Results of the form of (5.18) can also be obtained in some cases when σ2 is slowly varying
at zero by using the Theorems of Kono mentioned above or Theorem 6 in [M1]. Moreover,
in dealing with symmetric Levy processes, the constant C(y0) in (5.4) is independent of
y0, since by (5.14), the Gaussian processes associated with Levy processes are stationary.
Theorem 5.5 is not completely analagous to Theorem 2 in [Ba] because the latter result
is valid for all t ∈ R+ not just almost all t. Barlow’s Theorem 2 gives a uniform modulus
of continuity for the local time of Levy processes when σ2 is regularly varying with index
0 < α ≤ 1. We can not get this result by our methods. Our approach is to infer results
for local times from corresponding ones for Gaussian processes. But special properties of
the Gaussian processes associated with Levy processes seem to be involved in Theorem
2 in [Ba]. We say this because the analogue of Theorem 5.3 in [Ba] is false for Gaussian
processes. Let us be more precise. The approach of this paper is to use results for Gaussian
processes to obtain results for the local times of the associated Markov processes. The result
for Gaussian processes that might give Theorem 5.3 in [Ba] would be that σ(u)(2 log 1/u)1/2
28
is a lower uniform modulus of continuity for Gaussian processes for which σ satisfies the
hypotheses of [Ba, Theorem 5.3]. But this is not true as can be seen by the example in
[M]. Clearly, there are special properties of the Gaussian processes which are associated
with Levy processes that we would need to use to obtain Theorem 5.3 in [Ba]. So far we
have been unable to isolate these properties.
Exact uniform moduli of continuity are known for a wide class of Gaussian processes
with stationary increments. However, all the results we know of require that σ2 be concave
on [0, δ] for some δ > 0. This is not so great a restriction in the study of Gaussian processes
in genreral since for every such function σ2 one can find a stationary Gaussian procees, say
G(x), x ∈ R, such that E(G(x + h) − G(x))2 = σ2(h), for h ∈ [0, δ], (see e.g. Chapter
XV.3, Example b [Fe]). However, in general, it is hard to see whether the functions σ2
defined in (5.14) are concave on [0, δ] for some δ > 0. In fact, they are for symmetric stable
processes with index greater than 1. However, we will use a slightly different approach
to obtain exact uniform moduli of continuity for the local times of these processes. Of
course this is contained in [Ba, Theorem 2] but our proof is simple and illustrates a useful
technique of manipulating Gaussian processes. See also [Ba, Theorem 3].
Theorem 5.6. Let X be a symmetric stable process of index 1 < p ≤ 2. In particular,
let ψ(λ) = λp in (5.12). Let Lyt , (t, y) ∈ R+ × R be the local time of X. Then, for I a
closed interval of R
(5.19)
limδ→0
sup|x−y|≤δx,y∈I
|Lxt − Ly
t |(|x− y|p−1 log 1/|x− y|)1/2
= 2(Cp)1/2 supy∈I
(Lyt )1/2 for almost all t a.s.
where
(5.20) Cp =2π
∫ ∞
0
(1− cos y)yp
dy
Proof. The proof follows immediately from Theorem 1.4, which also holds for this kind of
modulus, (see the remarks following the proof of Theorem 4.5), and the fact that
(5.21) (2Cp|x− y|p−1 log 1/|x− y|)1/2
is an exact uniform modulus of continuity for the stationary Gaussian process Gp(x), x ∈I for which
σ2p(|x− y|) = E(Gp(x)−Gp(y))2 =
2π
∞∫0
1− cosλ(x− y)1 + λp
dλ
29
To see this we introduce two Gaussian processes with stationary increments, Gp,i(x), x ∈I, i = 1, 2, defined by
(5.22) Gp,i(x) =1√π
(∫ ∞
0
(1− cosλx)fi(λ) dB(λ) +∫ ∞
0
sinλxfi(λ) dB′(λ))
where
f1(λ) =1
λp/2and f2(λ) =
1(λp(1 + λp))1/2
and B and B′ are independent Brownian motions. Let Gp, Gp,1 and Gp,2 be independent
and note that Gp,1(x) and Gp(x) − Gp(0) + Gp,2(x) are equivalent Gaussian processes.
(They have the same covariance). Furthermore, we see by a change of variables that
E(Gp,1(x+ h)−Gp,1(x))2 =2π
∞∫0
1− cosλhλp
dλ = Cphp−1
It follows from [M1, Theorem 7] that (5.21) is an exact uniform modulus of continuity fro
Gp,1. (The Theorem just cited is writen for I = [0, 1]. It is easy to see, by a change of scale
that it is valid for any closed interval I). Therefore, (5.21) is an exact uniform modulus
of continuity for Gp(x) − Gp(0) + Gp,2(x). However, by Theorem 4.5 the term in Gp,2 is
“little o” of this modulus. Thus (5.21) is an exact uniform modulus of continuity for Gp,
which is what we wanted to show. This completes the proof of Theorem 5.6.
Using Theorem 5.1 we can obtain some information about the uniform modulus of
continuity of the local time of symmetric Levy processes that is not contained in [Ba].
Theorem 5.7. Let X be a symmetric Levy process and let Lyt , (t, y) ∈ R+ × R be the
for some δ′ > 0 where ρ is as given in Theorem 5.1 and also satisfies (5.2) and (5.3) and
σ is given in (5.14). Let φ(δ) be as defined in (5.10) and assume that (5.11) holds. Let I
be a closed interval of R. Then there exists a constant 0 < C <∞, independent of I, such
that
(5.23) limδ→0
sup|x−y|≤δx,y∈I
|Lxt − Ly
t |φ(|x− y|)
= C supy∈I
(Lyt )1/2 for almost all t a.s.
In particular if
ρ(δ) = (log 1/δ)−α α > 1/2
30
then
(5.24) limδ→0
sup|x−y|≤δx,y∈I
|Lxt − Ly
t |ρ(|x− y|)(log 1/|x− y|)1/2
= C supy∈I
(Lyt )1/2
for almost all t a.s.
and if
ρ(δ) = ((log 1/|δ|)(log log 1/|δ|)β)−1/2 β > 2
then
(5.25) limδ→0
sup|x−y|≤δx,y∈I
|Lxt − Ly
t |ρ(|x− y|)((log 1/|x− y|) log log 1/|x− y|)1/2
= C supy∈I
(Lyt )1/2
for almost all t a.s.
(The constants C are not necessarily the same).
Proof. Fix an interval I and let G(x), x ∈ I be a mean zero Gaussian process with
E(G(x)−G(y))2 = σ2(|x− y|). It follows from Theorem 4.5, (5.11) and the fact that φ(δ)
is monotone that
(5.26) limδ→0
sup|x−y|≤δx,y∈I
|G(x)−G(y)|φ(|x− y|)
≤ D1 a.s.
for some constant D1 <∞. Also the limit superior in (5.26) is greater than or equal to a
constant D2 > 0, since by Lemma 5.3 we get (5.5) with ψ replaced by φ and, of course, the
uniform modulus is greater than the local modulus. It now follows from Lemma 2.1 that the
limit superior in (5.26) is equal to a constant and it is clear, because of the homogeneity of
the condition on σ, that this constant is independent of I. We then get (5.23) by Theorem
1.4, which also holds for this kind of modulus, since the Gaussian processes associated with
Levy processes are stationary. (5.24) and (5.25) follow from (5.23).
In the above discussion of moduli of continuity we have confined our attention to the
real line because the 1–potential density of Levy processes on R2 is not finite. However,
there are examples of Markov processes on R2 with continuous local times, such as in the
recent work of Barlow and Bass [BB] on the Sierpinski carpet. All the results on moduli
of continuity in this Section can be extended to Rn whenever the local time is continuous.
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Michael B. Marcus Jay RosenDepartment of Mathematics Department of MathematicsThe City College of CUNY College of Staten Island, CUNYNew York, NY 10031 Staten Island, NY 10301
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